Adversarially guided diffusion sampling often achieves the target class, but sample quality degrades as deviations between the adversarially controlled and nominal trajectories accumulate. We formalize this degradation as a path-space Kullback-Leibler divergence(path-KL) between controlled and nominal (uncontrolled) diffusion processes, thereby showing via Girsanov's theorem that it exactly equals the control energy. Building on this stochastic optimal control (SOC) view, we theoretically establish that minimizing this path-KL simultaneously tightens upper bounds on both the 2-Wasserstein distance and Fréchet Inception Distance (FID), revealing a principled connection between adversarial control energy and perceptual fidelity. From a variational perspective, we derive a first-order optimality condition for the control: among all directions that yield the same classification gain, the component tangent to iso-(log-)density surfaces (i.e., orthogonal to the score) minimizes path-KL, whereas the normal component directly increases distributional drift. This leads to DPAC (Distribution-Preserving Adversarial Control), a diffusion guidance rule that projects adversarial gradients onto the tangent space defined by the generative score geometry. We further show that in discrete solvers, the tangent projection cancels the O(Δt) leading error term in the Wasserstein distance, achieving an O(Δt^2) quality gap; moreover, it remains second-order robust to score or metric approximation. Empirical studies on ImageNet-100 validate the theoretical predictions, confirming that DPAC achieves lower FID and estimated path-KL at matched attack success rates.

We introduce a novel reinforcement learning (RL) framework that treats distribution parameters as actions, redefining the boundary between agent and environment. This reparameterization makes the new action space continuous, regardless of the original action type (discrete, continuous, mixed, etc.). Under this new parameterization, we develop a generalized deterministic policy gradient estimator, Distribution Parameter Policy Gradient (DPPG), which has lower variance than the gradient in the original action space. Although learning the critic over distribution parameters poses new challenges, we introduce interpolated critic learning (ICL), a simple yet effective strategy to enhance learning, supported by insights from bandit settings. Building on TD3, a strong baseline for continuous control, we propose a practical DPPG-based actor-critic algorithm, Distribution Parameter Actor-Critic (DPAC). Empirically, DPAC outperforms TD3 in MuJoCo continuous control tasks from OpenAI Gym and DeepMind Control Suite, and demonstrates competitive performance on the same environments with discretized action spaces.